However, in practice, such rapid changes in density induces high interfacial energy between phases. To account for this, the Van der Waals-Cahn-Hilliard theory of phase transitions [1]} models the potential energy via an integral functional of the form
\(J_\varepsilon (u) := \int _\Omega W(u) + \varepsilon ^2\Vert \nabla u\Vert ^2\,dx, \qquad u \in W^{1,2}(\Omega ).\) 
